Results_uwnd_precip_Emily

Uwnd and precip
Emily Riley

2. Field1 means and variances: space, time, spacetime

1. The GRAND (space+time) mean of my x and y: -1.7987 m/s and 4.7379 (mm/d).
2. The GRAND (space+time) varianceof my x and y: 6.0236 (m/s)^2 and 12.6278 (mm/d)^2.
3. The GRAND standard deviations are: 2.4543, 3.5536 mm/d
4. The SPATIAL variance of my TIME MEAN longitude section is: 4.5413 (m/s)^2 and 5.7562 (mm/d)^2
5. The TEMPORAL variance of my LONGITUDE MEAN time series is: 0.07107 (m/s)^2 and 0.94054618 (mm/d)^2

3. (the assignment part)

1. Confirm that the time mean of the anomalies as defined above is 0. yes, bar_xprime = 0
2. Is the spatial mean of the anomalies (as defined above) 0?
   • No. The space-time mean is zero, though.
   • Is it the same as the time series of the spatial mean of the raw data? Or is it a new thing?
     • The anomaly time series for U-wind has a similar shape to the raw(x) data. The same broad features are there, like the dip in 1988, the double spike during 1997, and a dip centered around 2000. The amplitudes of the two time series are different. The difference (anomx - x) is a smooth piecewise function, though. The same is true for precip.
3. My CLIMATOLOGICAL ANNUAL CYCLE have variance:
   • Uwnd: 5.1552 (m/s)^2
   • 
   • Precip: 7.9339 (mm/d)^2
   • 
4. My INTERANNUAL ANOMALY ARRAYS have variance:
   • Uwnd: 0.8684 (m/s)^2
   • 
   • Precip: 4.6939 (mm/d)^2
   • 
5. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts.

	U1000	Precip
a) total variance of x	6.0236	12.6278
b) purely spatial (variance of TIME mean at each lon)	4.5413	5.7562
c) variance of (x minus its TIME mean at each lon)	1.4823	6.8716
d) purely temporal (variance of LON mean at each time)	0.07107	0.9405
e) variance of (x minus its LON mean at each time)	5.9525	11.6873
f) remove both means (space-time variability)	6.0236	12.6278
g) mean seasonal cycle	5.1552	7.9339
h) deseasonalized anomalies	0.8684	4.6939
i) variance of longitudinal mean of h	0.0352	0.2475
j) h minus i	0.8332	4.4465

6. Discuss your results:

• (a) = (g) + (h)
• Uwind: 6.0236 = 5.1552 + 0.8684
• Precip: 12.6278 = 7.9339 + 4.6939

_

4. (the assignment part)

1. Based on the variance_by_scalefactor diagram you make, what space and time scales (units: degrees and months) have an especially prominent lot of variance in your anomx field? These are the scales at which averaging over them reduces variance the most. Make a contour plot of anomx, or use Milan's total plots in Getting data -- can you see these "characteristic" scales by eye?
   • Most variance = small time and large space scales
   • As in Brian's plots, variance drops off faster in time rebinning vs. space rebinning.
   • Annotate the contour plot with some scale indications of about the right size (ovals in powerpoint may be easiest)
   • 
   • 

5. Scatter plot, correlation and covariance, regression-explained variance

1.

At first glance you can hint a relationship between wind and precipitation. Generally, when the wind is strongly eastward, precipitation is small, while larger precipitation values tend to occur when winds are westerly. On closer glance, though, there are differences between this relationship in the west and eastern hemisphere. In the central Pacific the above relationship holds, but not over the Eastern Pacific and Atlantic, where winds are always easterly. Plus, I get the sense that winds and precip are more correlated over ocean than land.

STRONGEST CORRELATION AT LONGITUDE 182.5:




2.

• The strongest correlation is at longitude 182.5:
  • rho = 0.767833
  • standard deviation of Uwnd = 1.7312340
  • standard deviation of Precipitation =3.4692004

• What fraction of the variance of y can be 'explained' by linear regression on x (y = mx + b)? How does this relate to rho? How much y variance is explained? (variance: with units of y squared) What is m?
  • m = rho*(stdevy/stdevx) = 0.383171
  • yvar_not_explained = 1 - rho^2 = 0.410433, so 0.589567 or 60% of the variance is accounted for by the regression.
  • 7.0956468 mm/d out of 12.035351 mm/d variance is explained.
  • The explained variance relates to rho, because the explained variance of y on x is the correlation coefficient squared times the variance of y. (I got help from Greta's HW here)
• What fraction of the variance of x can be 'explained' by linear regression on variable y? (x = nx + a)? How does this relate to rho? What is n?
  • n = rho*(stdevx/stdevy) = 1.5386515
  • The same fraction amount of variance is explained.
  • 1.7670334 m/s out of 2.9971711 m/s variance is explained.

3. Add Noise to y (Precip)

• Var(Preicp) = 12.035351 while Var(noise_Precip) = 13.696871Adding noise increased the variance by ~1.5
• The correlation decreased, 0.74994271
• 0.53304663 or 53% of the variance is explained.
• 6.6254788 of 12.429454 mm/d variance is explained.
• m = 0.80003573

---

6.

1. Show the zero-lag spatial covariance and correlation structures for your primary field. Interpret the results.

• The covariance for 1000 mb u-wind has two lobes of high covariance: 1) in the central Pacific near 180° and the other in the east Indian Ocean near 90°E. That 1000 mb u-wind covariance is high in the central Pacific is not surprising given ENSO effects on wind and precipitation are largest there. The high covariance in the east Indian Ocean may be linked to the Indian Ocean Dipole. Changes in the Indian Ocean SST gradient lead to pressure gradient and wind changes. Often times the Indian Ocean Dipole events co-occur with ENSO events, but not always.

• Correlation along the diagonal is generally 1 for Uwnd & itself and Precipitation & itself, as it should be. Not sure why a few places have correlation greater than one along the diagonal. The correlation drops below one fastest over land, i.e. the longitudes of South America ~300°, the Maritime Continent ~120°, Africa ~30°.
• The most interesting feature of the correlation plots, is that the correlation between wind and precipitation is highest in the central Pacific. Again, this indicates ENSO variability.

2. Show longitude-lag sections of the covariance or correlation of this field, for a base point at some longitude of interest.

• These plots are all for 1000 mb wind.

3. Intepret the results in terms of the characteristic space and time scales of your anomalies. Can you see these characteristic scales in your original raw data

• Zooming in on the bullseyes of covariance, we see that variance drops off faster in time than in longitude, which confirms our results from part one. But after the initial quick drop off in time, the covariance seems to decrease about the same in time and space (at least according to these plots).

4. Share a longitude-lag slice of your lagged co-variance matrix for your TWO fields. Label it, interpret it.

• The covariance of 1000 mb wind and precipitation is largest in the central Pacific (as shown earlier) at 182.5°. Here variance seems to drop off equally in time and space. The tilt in the variance indicates that prior to ENSO events the most variance between wind and precipitation is in the East Pacific. The variance shifts to the central and then west Pacific during an ENSO cycle. (Not quite sure I'm interpreting this correctly, but a negative lag means prior to zero lag.)